# Properties

 Label 4005.2.a.p Level 4005 Weight 2 Character orbit 4005.a Self dual Yes Analytic conductor 31.980 Analytic rank 1 Dimension 8 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4005 = 3^{2} \cdot 5 \cdot 89$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4005.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$31.9800860095$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{4} + q^{5} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{7} + ( -\beta_{1} - \beta_{2} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{4} + q^{5} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{7} + ( -\beta_{1} - \beta_{2} ) q^{8} -\beta_{1} q^{10} + ( -2 + \beta_{2} ) q^{11} + ( -2 + \beta_{1} + \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{13} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} ) q^{14} + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{16} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{17} + ( 3 - 2 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{19} + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{20} + ( 4 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{22} + ( 1 + \beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{23} + q^{25} + ( 1 - \beta_{1} - 4 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{26} + ( -5 + 5 \beta_{1} + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{28} + ( -2 + 2 \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{29} + ( \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{31} + ( -1 - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{32} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{34} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{35} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{37} + ( 3 - 2 \beta_{1} + \beta_{3} - 2 \beta_{6} - 2 \beta_{7} ) q^{38} + ( -\beta_{1} - \beta_{2} ) q^{40} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{41} + ( -1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + \beta_{7} ) q^{43} + ( -1 + 5 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{44} + ( -3 + 3 \beta_{1} + \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{46} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{47} + ( 1 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{7} ) q^{49} -\beta_{1} q^{50} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{52} + ( 2 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{53} + ( -2 + \beta_{2} ) q^{55} + ( 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{56} + ( -5 + 5 \beta_{1} + \beta_{2} + 2 \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{58} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{59} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{61} + ( -4 + 6 \beta_{1} - \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{62} + ( -\beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{64} + ( -2 + \beta_{1} + \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{65} + ( -3 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 4 \beta_{6} ) q^{67} + ( -5 + 7 \beta_{1} + \beta_{2} + 6 \beta_{3} + 5 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{68} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} ) q^{70} + ( -4 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{71} + ( -1 - \beta_{1} - 3 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{73} + ( -1 + \beta_{1} + 2 \beta_{3} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{74} + ( 3 - 7 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{76} + ( -8 \beta_{1} - \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{77} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - 4 \beta_{5} - 2 \beta_{7} ) q^{79} + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{80} + ( 1 - \beta_{1} - 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{7} ) q^{82} + ( -6 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{7} ) q^{83} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{85} + ( 2 - 2 \beta_{1} - 4 \beta_{2} + \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{86} + ( -11 + 6 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{88} - q^{89} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{91} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{92} + ( -6 + 7 \beta_{1} + 5 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{94} + ( 3 - 2 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{95} + ( 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{97} + ( 5 - 8 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - q^{2} + 7q^{4} + 8q^{5} - 6q^{7} - 3q^{8} + O(q^{10})$$ $$8q - q^{2} + 7q^{4} + 8q^{5} - 6q^{7} - 3q^{8} - q^{10} - 14q^{11} - 7q^{13} - 15q^{14} + 9q^{16} - 17q^{17} + 17q^{19} + 7q^{20} + 2q^{22} + q^{23} + 8q^{25} - 3q^{26} - 29q^{28} - 10q^{29} + q^{31} - 2q^{32} - 16q^{34} - 6q^{35} - 11q^{37} + 30q^{38} - 3q^{40} - 15q^{41} - 5q^{43} - 7q^{44} - 12q^{46} - 12q^{47} + 4q^{49} - q^{50} - 14q^{52} + q^{53} - 14q^{55} - 3q^{56} - 37q^{58} - 26q^{59} + 13q^{61} - 22q^{62} - 15q^{64} - 7q^{65} - 25q^{67} - 23q^{68} - 15q^{70} - 28q^{71} - 17q^{73} + 5q^{74} + 8q^{76} - 7q^{79} + 9q^{80} + 5q^{82} - 44q^{83} - 17q^{85} + 13q^{86} - 66q^{88} - 8q^{89} + 27q^{91} - 15q^{92} - 27q^{94} + 17q^{95} + q^{97} + 34q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 11 x^{6} + 9 x^{5} + 34 x^{4} - 19 x^{3} - 27 x^{2} + 11 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 5 \nu$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{7} + \nu^{6} + 23 \nu^{5} - 8 \nu^{4} - 76 \nu^{3} + 12 \nu^{2} + 65 \nu - 7$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$-\nu^{7} + \nu^{6} + 11 \nu^{5} - 9 \nu^{4} - 34 \nu^{3} + 18 \nu^{2} + 27 \nu - 7$$ $$\beta_{5}$$ $$=$$ $$($$$$-4 \nu^{7} + 3 \nu^{6} + 45 \nu^{5} - 26 \nu^{4} - 144 \nu^{3} + 50 \nu^{2} + 121 \nu - 27$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{7} - 4 \nu^{6} - 55 \nu^{5} + 34 \nu^{4} + 170 \nu^{3} - 61 \nu^{2} - 136 \nu + 27$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$-6 \nu^{7} + 5 \nu^{6} + 67 \nu^{5} - 42 \nu^{4} - 212 \nu^{3} + 72 \nu^{2} + 175 \nu - 29$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} - \beta_{3} - \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 6 \beta_{5} - 8 \beta_{4} - 7 \beta_{3} - 7 \beta_{1} + 15$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} + 2 \beta_{6} + \beta_{5} + 8 \beta_{2} + 28 \beta_{1} + 1$$ $$\nu^{6}$$ $$=$$ $$11 \beta_{7} + 2 \beta_{6} + 37 \beta_{5} - 54 \beta_{4} - 48 \beta_{3} - 47 \beta_{1} + 86$$ $$\nu^{7}$$ $$=$$ $$13 \beta_{7} + 24 \beta_{6} + 12 \beta_{5} - \beta_{4} - 3 \beta_{3} + 54 \beta_{2} + 163 \beta_{1} + 9$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.51468 2.23321 1.16343 0.217002 0.151894 −1.11667 −1.66289 −2.50065
−2.51468 0 4.32363 1.00000 0 −2.83060 −5.84320 0 −2.51468
1.2 −2.23321 0 2.98721 1.00000 0 1.87039 −2.20463 0 −2.23321
1.3 −1.16343 0 −0.646427 1.00000 0 2.23155 3.07894 0 −1.16343
1.4 −0.217002 0 −1.95291 1.00000 0 −2.89958 0.857790 0 −0.217002
1.5 −0.151894 0 −1.97693 1.00000 0 2.88745 0.604071 0 −0.151894
1.6 1.11667 0 −0.753041 1.00000 0 −1.16728 −3.07425 0 1.11667
1.7 1.66289 0 0.765209 1.00000 0 −1.19579 −2.05332 0 1.66289
1.8 2.50065 0 4.25326 1.00000 0 −4.89614 5.63461 0 2.50065
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$89$$ $$1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4005))$$:

 $$T_{2}^{8} + \cdots$$ $$T_{7}^{8} + \cdots$$ $$T_{11}^{8} + \cdots$$